Is there an undirected tree graph with all vertices with the same degree?

Question

I'm trying to draw an undirected tree graph such that all vertices have the same degree, but it seems that I can't.

Is it possible? I'd love an example. If it's not possible, what am I missing?

Answer 1

A tree on $n\geq 1$ vertices has $n-1$ edges. Therefore, if all the vertices have the same degree $d\geq 0$ then $$dn=\sum_{v\in V}\text{deg}(v)=2(n-1)$$ that is $(2-d)n=2$ which has two solutions: i) $d=1$ and $n=2$; ii) $d=0$ and $n=1$.

What may we conclude?

Answer 2

It is a known fact that every tree has at least one leaf, i.e. a vertex of degree 1 (unless of course the tree is trivial, in which case you have only one vertex and its degree is 0). To see this, take a path P of maximum length. If the last vertex of P has degree strictly more than 1, then one its neighbours would have to lie on P (otherwise P would not have maximum length), but that would form a cycle which is not possible.

So in every tree there is a vertex of degree 1. This means that in order to have a tree where all vertices have the same degree, that degree would have to be exactly 1. The only way this would be possible would be if the tree had exactly 2 vertices.